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Downloads.Php] Strazewski Journal of Systems Chemistry 2010, 1:11 http://www.jsystchem.com/content/1/1/11 PERSPECTIVES Open Access The relationship between difference and ratio and a proposal: Equivalence of temperature and time, and the first spontaneous symmetry breaking Peter Strazewski Abstract The mathematical form of the energy-entropy relationship is that of the relationship between the difference and the ratio of any two ‘entities’ x and y creating a geometrical 3D projection x = y · z, i.e., a surface of the shape of a hyperbolic paraboloid being a particular form of a quadric. The significance of this relationship is discussed here in a realm beyond thermodynamics. Somewhat intuitively, yet still in strict mathematical analogy to the relation between its most fundamental state variables, I propose for any isolated universe the cosmological state variables, absolute temperature and absolute time, to be equivalent much in the same way as energy and entropy are equivalent for an isolated ensemble of similar objects. According to this principle, the cosmological constant is inversely proportional to the squared product of absolute time and absolute temperature. The spontaneous symmetry breaking into a time component and a temperature component of the universe takes place when the first de facto irreversible movement occurs owing to a growing accessed total volume. For any element aa there is an inverse element –1 : The relationship between difference and ratio in (G3) mathematics and physics aa ––1 = a11 ae= In mathematics the concept of ‘agroup’ is used to for- mulate, in a most general way, any kind of mathematical Any pair of elements ab and are commutative : operation, for example, addition, multiplication or rota- (G4) = tion. Generally, a mathematical group consists of ‘ele- ab ba ’ ments a, b, c, ... and an instruction that attributes to The mathematical operationofadditioninconjunc- each pair of elements, say a and b, a new element ab tion with elements that are integers, {a, b, c, ...} Î ℤ,is in such a way that three group axioms G1, G2 and G3 an abelian group. To specify, replace the above general strictly apply. Abelian groups are commutative for formalism with a specific one, i.e., replace with +, e – which group axiom G4 applies as well: with 0, and a 1 with –a. The mathematical operation of For any three elements ab,: and c associativity applies multiplication in conjunction with elements that (G1) are positive real numbers, {a, b, c,...}Î R+,isalso (()ab ca= () bc − an abelian group: :,:=⋅e =1 , and aa1 :/=1 . Note that, whereas the addition of real numbers is an abelian For any element ae there is an identity element : group, axiom G3 usually does not apply to the multipli- (G2) + ea== ae a cation of positive integers {a, b, c, ...} Î ℤ because gen- – – – erally {a 1, b 1, c 1,...}∉ ℤ. Most commonly, mathematical groups are used to describe geometrical symmetries; the vast majority of mathematical groups are non-abelian (axiom G4 does not apply), in particu- Correspondence: [email protected] lar, when the parameter space is higher than 2D (two- Laboratoire de Synthèse de Biomolécules, Institut de Chimie et Biochimie Moléculaires et Supramoléculaires (CNRS UMR 5246), Université Claude dimensional). Bernard Lyon 1, Université de Lyon, 43 bvd du 11 novembre 1918, F - 69622, Villeurbanne, France © 2010 Strazewski; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Strazewski Journal of Systems Chemistry 2010, 1:11 Page 2 of 7 http://www.jsystchem.com/content/1/1/11 In physics and physical cosmology this formalism is generalised fashion the relationship between difference u extremely useful for the description of spontaneous and ratio x/y as depicted in Figure 1. Note that v is the symmetry breaking into different fundamental forces root of the function in equation 2. through the application of gauge theories, such as in To analyse more readily, a set of 2D projections of the Quantum Electrodynamics (QED) or Quantum Chromo- same function is depicted in Figure 2 where u is dynamics (QCD) being the basis of the Standard Model. sampled over a representative range of numerical values A gauge theory is a type of field theory in which the for x and y. Note that, trivially, x/y requires for v >0 Lagrangian is invariant under a certain continuous the root to lie in the quadrants where both x and y have group of local transformations. The Lagrangian of a thesamesign(righthalfof the 2D plot), whereas a dynamical system is a function that summarises the negative value for v places the root in the quadrants of dynamics of the system. Common to all these theories opposite signs for x and y (reflection of the hyperbles is the utilization of Legendre transformations. The through x/y = 0, not shown). Legendre transformation can be generalized to the In special relativity the relationship between difference Legendre-Fenchel transformation. It is commonly used and ratio immediately visualises how the shape of in thermodynamics and in the Hamiltonian formulation squared distance differentials in spacetime ds2 (ordinate of classical mechanics, for example, to describe various in the above 2D plot) relates to the ratio of squared thermodynamic potentials in classical thermodynamics, spatial distance differentials over squared time differen- 2 2 2 to derive partition functions from a state equation in sta- tials d/dxti for v =c (abscissa). In thermodynamics tistical thermodynamics, to link Lagrange (classical) the above relationship shows how changes in Gibbs free mechanics with Hamiltonian (quantum) mechanics, to energy ΔGT relate to the temperature at which ΔG formulate QED and QCD, and more. In a thermody- vanishes, T = ΔH/ΔS for ΔCp = 0, or in a differential namic context the linking of energy with entropy through form for infinitesimal state changes: dGT vs. T =dH/dS. a quantified similarity argument, as described in group It was shown that the ΔH and ΔS values of similar thermodynamics [1], may be viewed as a Legendre trans- objects at their respective equilibrium temperatures are formation on a group, rather than individual, level. all linearly related to their corresponding ΔGT values Consider a mathematically general relationship [1]. These realised experimental values of similar objects between difference and ratio (quotient) being the opera- gather linearly, in the above plot, close to the root of tional results obtained from, respectively, the substrac- the function at {x – v · y =0,x/y = v}, i.e., close to the tion and division of the same elements, say, x and y saddle point. Physics imposes positive v values in both (eqns. 1 and 2): special-relativity and thermodynamics, since there is no immediate physical meaning for a negative speed of ux=−⋅ y (1) light (reaching in vacuo a maximum constant positive value c) nor negative absolute temperature, both, in xy//()=⋅ x x − u (2) accord with the Second Law. General and special relativity describe the shape of Interpret x and y as any two parameters that are and the dynamics within a spacetime where the objects needed to define the characteristics of some state and may or may not move in a reversible fashion. The main v as a mathematically independent variable. For exam- focus in relativity theory is not reversibility but rather to ple, {x, y} may be the coordinates of a point in 2D space describe objects (particles) and their ‘directed’ move- and v a force acting on one coordinate (in one direc- ments at up to relativistic speeds along geodesics in tion) but not the other. If x were to represent the spacetime being shaped by gravitational fields. Thermo- distance between two objects in Euclidian 3D space, dynamics describes the energetics of dynamic systems difference u would describe the movement of these that contain multi-component objects (‘macrostates’) objects through spacetime according to Einstein’s special able to adopt, essentially reversibly, multiple isoergonic 2 =−⋅2 22 relativity ddcdsxi twhere d = differential, s = (degenerate) ‘microstates’. Both theories are, in terms of distance in spacetime, xi = spatial Cartesian coordinates physics, quite different from one another. The former is {x, y, z}, c = vacuum speed of light, and t = time. Differ- based on Einstein’s mass-energy equivalence; in the lat- ence u could also represent the incremental change Δ in ter an energy-entropy equivalence of macrostates was free energy upon transformation of one (metastable) identified within groups of similar objects. Mathemati- macrostate into another, as expressed in the Gibbs- cally, the former is generated from applying Pythagorean ΔΔ=−⋅ Δ Δ Helmholtz equation GHTST ,where GT = rules to the coordinates of a 4D Lorentzian geometry GibbsfreeenergychangeattemperatureT, ΔH = (two sums and a difference between four positive terms: energy change at constant pressure (enthalpy change), 2 ++−⋅2 2 22 dddcdxxx1 2 3 t). The latter originates from and ΔS = entropy change. Equation 2 describes in a correlating and partitioning energy contributions Strazewski Journal of Systems Chemistry 2010, 1:11 Page 3 of 7 http://www.jsystchem.com/content/1/1/11 Figure 1 3D view of the relationship between difference and ratio. Equation 2 at v = 300. The saddle point of this hyperbolic paraboloid is {u = x – v · y =0,x =0,x/y = 300}. through Legendre transformations. In spite of this the assumption, unproven as it is, that the energy- apparent disparity, the mathematical formalism looks entropy relationship for an ensemble of similar objects alike for both theories. Lorentzian geometry and the [1] applies to the energetics of all physical changes Gibbs-Helmholtz equation are both typified through a irrespective of scale and material; actually, irrespective crucial substraction term; through the mathematical dif- of whether it is applied to normal matter, dark matter, ference viz.
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